Gov. Cuomo, fearing a clash with a Mayor Bill de Blasio over the poll-leading Democrat’s tax-the-rich plan to fund a universal pre-K program, has directed his staff to explore “alternative options’’ to let the city raise the money without an income-tax hike, The Post has learned.

“The last thing the governor wants is a clash with a Mayor de Blasio,’’ a Cuomo administration source said.

“He has his staff looking at the options. There are things that the state could do other than permitting the city to raise the income tax that could make this [de Blasio’s campaign pledge] possible,’’ the source said.

De Blasio has vowed to fund his full-time universal pre-kindergarten plan with a tax boost of more than $500 million annually on those making $500,000 or more a year.

At the same time, Cuomo has pledged to cut taxes next year — when he’ll be seeking re-election.

Administration sources said the seemingly opposite tacks could be harmonized.

“The governor wants tax cuts for next year, which obviously suggests that there’s going to be some form of budget surplus in the current fiscal year, and if there’s a surplus, that means there could be money for spending increases, such as money for a universal pre-K program,’’ said a source.

Another possibility for Cuomo to help fund deBlasio’s campaign promise: state authorization to increase several smaller city taxes that are largely paid for by the well-to-do.

Sources said the governor is determined to prevent de Blasio from imposing a new tax on the rich, because Cuomo has already imposed his own “millionaire’s tax’’ after vowing not to do so.

Cuomo also intends to avoid a head-on clash with de Blasio and the “progressive,’’ or left-of-center, wing of the Democratic Party that de Blasio represents, since they could be crucial if the governor seeks to run for president in 2016.

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Cuomo’s aides and Health Department officials were privately using words like “disaster’’ and “chaos’’ to describe last week’s ObamaCare launch in New York — even as aides to Cuomo publicly put a happy face on the massive computer glitches and failures that marked the debut.

While department spokeswoman Donna Frescatore claimed “more 44,000 people have visited and actively shopped’’ for health insurance on the state’s ObamaCare Web site, she conspicuously failed to say how many had actually purchased an insurance policy.

By the end of the week, the department was refusing to disclose that number, although one insider with firsthand knowledge said, “It’s only a few thousand.’’

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Two actions by Cuomo left longtime observers shaking their heads in disbelief last week as he named former Gov. George Pataki and former state Comptroller Carl McCall — rivals he ripped as unfit when he ran for governor in 2002 — to head a tax-reduction-study commission and claimed dysfunctional Washington should look to Albany as a model.

A knowledgeable administration source also called the governor’s oft-made claim that, in contrast with Washington, state government is running smoothly “a myth,’’ noting, “The governor has presented this image that things are efficient and locked down, but when you pull the curtain back, it’s worse than sausage-making.

“It’s just that the bar was so low after [Govs. Eliot] Spitzer and [David] Paterson that any semblance of competency is greeted with approval.’’

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Things can get pretty petty in Albany: When The Post disclosed at the end of the summer that Cuomo’s agriculture commissioner, Darrell Aubertine, would resign in September, Cuomo’s aides claimed the report was false — and ordered Aubertine to wait until last Friday.

“They didn’t want The Post to be right, so they had him wait until the start of October,’’ said one administration source.